On dual-containing, almost dual-containing matrix-product codes and related quantum codes

IF 1.2 3区 数学 Q1 MATHEMATICS
Meng Cao
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引用次数: 0

Abstract

Matrix-product (MP) codes are a type of long codes formed by combining several commensurate constituent codes with a defining matrix. In this paper, we study the MP code when the defining matrix A satisfies the condition that AA is (D,τ)-monomial. We give an explicit formula for calculating the dimension of the hull of a MP code. We present the necessary and sufficient conditions for a MP code to be dual-containing (DC), almost dual-containing (ADC), self-orthogonal (SO) and almost self-orthogonal (ASO), respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code that is DC, ADC, SO and ASO, respectively. We give alternative necessary and sufficient conditions for a MP code to be ADC and ASO, respectively, and show several cases where a MP code is not ADC or ASO. We give the construction methods of DC and ADC MP codes, including those with optimal minimum distance lower bounds. We introduce the notation of τ-optimal defining (τ-OD) matrices and provide the criteria for determining whether two types of k×k matrices are τ-OD matrices at k=3 and k=4, respectively. We give many examples of DC and ADC MP codes involving τ-OD matrices, some of which are optimal or almost optimal according to the Database [11]. By applying the generalized Steane's enlargement procedure to these DC MP codes, we obtain some good quantum codes that improve those available in the Database [7].

论双含、近双含矩阵积码及相关量子码
矩阵-乘积(MP)码是一种长码,由多个相称的组成码与一个定义矩阵组合而成。本文研究了定义矩阵 A 满足 AA⊤ 是 (D,τ) 单项式这一条件时的 MP 码。我们给出了一个明确的公式来计算 MP 代码的壳维度。我们分别提出了 MP 码为含对偶码(DC)、近似含对偶码(ADC)、自正交码(SO)和近似自正交码(ASO)的必要条件和充分条件。我们从理论上确定了涉及组成代码之间关系的所有可能方式的数量,这些方式分别产生了 DC、ADC、SO 和 ASO 的 MP 代码。我们给出了 MP 编码分别是 ADC 和 ASO 的其他必要条件和充分条件,并展示了 MP 编码不是 ADC 或 ASO 的几种情况。我们给出了 DC 和 ADC MP 编码的构造方法,包括具有最优最小距离下界的编码。我们介绍了τ-最优定义(τ-OD)矩阵的符号,并提供了在 k=3 和 k=4 时判断两种 k×k 矩阵是否为τ-OD 矩阵的标准。我们给出了许多涉及 τ-OD 矩阵的 DC 和 ADC MP 编码的例子,根据数据库[11],其中一些是最优或接近最优的。通过对这些直流 MP 代码应用广义的 Steane 放大程序,我们得到了一些很好的量子代码,这些代码改进了数据库 [7] 中的代码。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.00
自引率
20.00%
发文量
133
审稿时长
6-12 weeks
期刊介绍: Finite Fields and Their Applications is a peer-reviewed technical journal publishing papers in finite field theory as well as in applications of finite fields. As a result of applications in a wide variety of areas, finite fields are increasingly important in several areas of mathematics, including linear and abstract algebra, number theory and algebraic geometry, as well as in computer science, statistics, information theory, and engineering. For cohesion, and because so many applications rely on various theoretical properties of finite fields, it is essential that there be a core of high-quality papers on theoretical aspects. In addition, since much of the vitality of the area comes from computational problems, the journal publishes papers on computational aspects of finite fields as well as on algorithms and complexity of finite field-related methods. The journal also publishes papers in various applications including, but not limited to, algebraic coding theory, cryptology, combinatorial design theory, pseudorandom number generation, and linear recurring sequences. There are other areas of application to be included, but the important point is that finite fields play a nontrivial role in the theory, application, or algorithm.
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